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Abstract

Background

We examine the effect of heat waves on mortality, over and above what would be predicted
on the basis of temperature alone.

Methods

Present modeling approaches may not fully capture extra effects relating to heat wave
duration, possibly because the mechanisms of action and the population at risk are
different under more extreme conditions. Modeling such extra effects can be achieved
using the commonly left-out effect-modification between the lags of temperature in
distributed lag models.

Results

Using data from Stockholm, Sweden, and a variety of modeling approaches, we found
that heat wave effects amount to a stable and statistically significant 8.1-11.6%
increase in excess deaths per heat wave day. The effects explicitly relating to heat
wave duration (2.0–3.9% excess deaths per day) were more sensitive to the degrees
of freedom allowed for in the overall temperature-mortality relationship. However,
allowing for a very large number of degrees of freedom indicated over-fitting the
overall temperature-mortality relationship.

Conclusions

Modeling additional heat wave effects, e.g. between lag effect-modification, can give
a better description of the effects from extreme temperatures, particularly in the
non-elderly population. We speculate that it is biologically plausible to differentiate
effects from heat and heat wave duration.

Background

Heat stress can lead to fatal consequences due to: dehydration; increased cardiovascular
stress; kidney dysfunction; and electrolyte disorders [1,2]. At a population level, many studies show mortality tends to rise with higher temperatures
[3]. Two approaches are generally used to quantify excess mortality: studies that focus
exclusively on heat waves (so called episode studies); and studies that use time series
to estimate the effects of temperature on mortality by averaging over hot days and
heat waves. Heat waves are commonly referred to as a period of extreme heat stress
relative to the normal climate, although the exact definition varies according to
the number of consecutive days of heat, temperature variable(s) and heat threshold.
Many time series studies, assuming the association between temperature and mortality
is non-linear, report associations between heat and mortality that are immediate or
delayed by up to a week [4,5]. However, the validity of this approach is challenged by research that reports an
additional effect for heat waves [6]. Other studies have reported that heat-related mortality is sensitive to the duration
of heat waves regardless of the intensity of the ambient heat (e.g. in France during
the 2003 heat wave [7]). A number of studies have since explored additional heat waves effects with respect
to their timing, intensity, duration and location [6,8-13]. All studies found statistically significant additional risks that may relate to
the duration of heat waves and the cumulative extreme heat exposure. The main differences
between the studies were the models used to estimate the heat-mortality relationship
and the location.

The reason why the overall temperature-mortality relationship may not fully explain
effects during heat waves is because the physiological effects of high temperatures
and heat waves are different. For example, cumulative heat stress during a prolonged
heat wave is more likely to cause dehydration. Cumulative heat stress is also more
strongly related to cardiovascular deaths [8,13]. Differences in age stratified relative risks to heat and to heat waves have shown
that the population at risk may differ, with the middle aged population potentially
at the highest risk during heat waves [8,13].

We explain why additional heat wave effects are not perfectly captured in models of
temperature-related mortality, and illustrate the estimation of additional heat wave
effects using empirical data. We also explore how the additional heat wave effect
is dependent on the complexity of temperature-mortality model. Using prior studies
and our own results, we argue that future studies should evaluate potential additional
effects from heat waves by decomposing heat exposure into a temperature term and an
added heat wave term. Increasing model complexity will not suffice if there is no
differentiation between heat wave days and non-heat wave days.

Modelling weather-related mortality

Time series methods based on daily data have been developed and applied in studies
of the short-term health effects of environmental factors like air pollution and weather
[14,15]. Models often include lagged effects of exposure and adjustment for potential mortality
displacement [14,16,17]. Studies have also explored: the use of non-linear functions to adjust for confounding
(e.g., season), allowing non-linear exposure-response relationships, and the use of
model fitting criteria [17,18].

Why additional heat wave effects?

Additional heat wave effects may appear as an artefact if models employ an overly
simple exposure-response relationship that does not adequately capture the non-linear
effect with higher temperatures. In this case the additional effect is related directly
to the potential misspecification of the effect. This is illustrated in Figure 1 where a linear relationship and a non-linear relationship are fitted to explain the
relative risk (RR). Assuming that the factual relationship is wrongly assumed linear,
an additional heat wave effects would try to compensate the difference between these
two curves. This type of misspecifications can be addressed by allowing a more flexible
non-linear exposure response relationship.

Figure 1.The fit of a linear (black) and a non-linear (grey) curve to data.

Additional heat wave effects can also arise due to cumulative heat stress. It is tempting
to believe that cumulative stress can be estimated just like other delayed effects
of heat exposure effects, e.g. by distributed lag models. This is not the case. Distributed
lag models allow temperature a few days or weeks prior to day t to affect mortality on day t, and so are perfectly able to capture delayed effects of temperature. However, distributed
lag models assume the delayed effects are related to the temperature on day t only. This means that delayed lag effects are independent of other lag days, for
example, days t-1 and t-2. They cannot model health effects caused by the temperature being above a heat threshold
for a number of consecutive days. Thus, the distributed lag effect at a certain day
of the heat wave does not estimate the effects relating to persistent heat stress.
In order to estimate the effects relating to several days consecutive heat exposure
above a certain threshold one would need to include non-linear interactions (effect-modifications)
between the temperature lag effects. Here the non-linearity relates to the fact that
the additional heat wave effects are thought to appear above some extreme temperature
threshold. Thus, short-term cumulative stress of extreme heat can be described by
lag effect-modification.

For illustration we consider the simple case where the effects of temperature are
assumed to be dichotomous according to a threshold of the 98th percentile of the temperature distribution.

(1)

The effect of extreme temperature (T) on mortality up to lag 3 is then in a non-constrained
distributed lag model be estimated in a regression model:

(2)

(3)

This model assumes that the effect of extreme temperature on day t is independent of whether the temperatures on day t–1 and t–2 are also extreme. This would not be the case if there are additional effects from
persistent periods of extreme heat. Then, mortality on day t is also conditional on the temperatures on day t–1 and t–2. To capture this in a regression model we can add two- and three-way interactions
of the lag terms. In the model below we assume there are additional conditional relationships
only between lags that are consecutive in time (e.g. an extended period without relief
from the extreme temperatures),

(4)

When the number of lags studied is large (or if heat waves are long) this type of
model will require a large number of interaction terms to be estimated, resulting
in collinearity (similarly to the unconstrained distributed lag model). To avoid this
we can add all the interaction terms together to create one variable denoting extended
heat wave periods. This binary indicator variable is set to 1 if at least two or more
proceeding days exceed the extreme temperature threshold, and 0 otherwise. We refer
to this variable as a heat wave indicator variable (HWI). Note that our HWI is equal
to Tt× Tt-1, but it cannot be separated from days where . Thus, the HWI variable describes the two-way interaction that was previously described
in model (b). A new model including this variable can be expressed as,

(5)

The model (c) with the HWI will not differentiate between two and three way interactions
(e.g. two and three days of heat wave), as it assumes the additional effect is the
same independent of the length of the heat wave. We might, however, have reason to
suspect that longer heat waves are associated with larger additional effects. In this
case we can construct a new variable with distinct values for shorter and longer heat
waves. We refer to this variable as a heat wave duration variable (HWD). In the case
where we use only 3 lag days we define HWD as to equal . In this example the HWD variable takes the values 0, 1, and 2 depending on how many
consecutive days of extreme temperatures there are in before day t. The effect of heat wave duration can be estimated as a linear or non-linear function,
or as a factor variable directly yielding the interactions. A non-linear function
would be sensible if, for example, the effect of duration increased quickly and then
remained high. Assuming linearity we get the regression model:

(6)

The model (d) estimates the effects of the duration of heat waves as a linear function
avoiding potential collinearity induced by explicitly including many lag interaction
variables. The models above assume a dichotomous exposure-response relationship for
temperature, and that effects of temperature and heat waves are not extended over
more than 3 lag days. However, the same principle can be used when fitting a distributed
lag model and a larger number of lag days.

Model choice and additional heat wave effects

While recent studies found significant additional effects from heat waves, particularly
in cold climates (such as the northern parts of the US and Sweden [9,19]), the size of this effect has been estimated using different approaches [9,10,12,13]. The main differences were: i) the complexity of the model used for the exposure-response
relationship between temperature and mortality; ii) allowing for spatial heterogeneity
in the additional heat wave effect; iii) allowing for heterogeneity in the additional
heat wave effects between population sub-groups.

The study presenting the smallest additional effect from heat waves found the size
of the effects to be almost negligible when the exposure-response relationship was
allowed a very flexible parameterization using two dimensional cubic spline functions
for temperature and lag day and modeling the main effect in a first stage model and
the effect-modification in a second stage [12]. However, the estimates were for all-cause mortality independent of geographical
differences in U.S. cities, while there is evidence that heat wave effects may be
very sensitive to age, cause of death and location [8,9,13,19]. In particular, additional heat wave effects were negligible in the southern US,
and large in the north east of the US [1,9,19].

Two studies estimated the overall temperature-mortality relationship using non-linear
distributed lags and two-dimensional spline functions, and then tested the sensitivity
of this parameterization [10,20]. Other studies used less complex linear and/or non-linear exposure response relationships,
with a small number of lag days of between 1 and 3 [6,8,9,13,21]. The lag days in these studies were chosen according to prior literature and model
fit criteria.

Bobb et al. argued against fitting one model across a range of climates (using the
same degrees of freedom, splines and temperature measures), as they found that in
most cities there were two or more models with a similar fit to the data [19]. Interestingly, after averaging over many different models they found larger effects
of temperatures on heat wave days compared to non-heat wave days [19].

Estimation of additional heat wave effects in Stockholm, Sweden

Methods

We applied a non-linear distributed lag model to the effects of temperature in Stockholm
County, Sweden, on total mortality during the years 1990–2002. Table 1 describes the basic characteristics of the data. We used daily maximum temperature
as the predictor of daily mortality, adjusting for long-term time trends, seasonality,
days of the week and national holidays in a non-linear distributed lag framework.
This is model (1).

Table 1.Descriptive statistics for daily deaths and environmental variables in Stockholm County,
1990–2002, per season

To estimate additional heat wave effects (e.g. lag effect-modifications) we incorporated
the variables HWI and HWD as described in a previous section. The frequencies of the
HWD and HWI variables are in Table 2. Heat waves were defined as at least two days with maximum daily temperature above
the 98th percentile. So, on the second day with a temperature over the threshold, the variable
was 1, and on the 7th consecutive day the variable was 6. The first day of temperature
above the threshold was set to 0 corresponding to no accumulated heat effects. We
first fitted the effect of heat wave duration as a smooth function (penalized spline
with 4 degrees of freedom), but as it showed an approximately linear association we
fitted a linear term. The model with the parameter for duration of heat waves, HWD,
is model (2). The duration term estimates the effect-modification of cumulative lag
terms above the 98th percentile indirectly, with the prior assumption that the length of the heat wave
period can influence the mortality response.

Table 2.Values and frequencies taken by the heat wave duration and the heat wave indicator
variable

In model (3) we estimated the additional effects from heat waves (maximum temperature
above 98th percentile for at least two days) using an indicator variable, HWI. This model does
not estimate additional effects due to heat wave duration explicitly, but the average
additional excess mortality during the heat wave periods. The heat wave indicator
estimates the effect modification of lag terms above the 98th percentile assuming all days above this threshold are equally contributing to the
mortality response.

As equations the three models are:

(7)

(8)

(9)

(10)

Here t is the time in days, S is a cubic spline function, the spline function of temperature is two dimensional
with lag degree of freedom given by lag.df and variable degree of freedom given by var.df. HWD is a linear variable denoting the day of the heat waves, HWI is an indicator
variable for heat waves. timet estimates trends and seasonal changes using a spline with 6 degrees of freedom per
year (78 degrees of freedom in total), DOW denotes the day of week, and HD denotes national holidays.

We used the “dnlm” package in R [22]. We tested degrees of freedom for temperature from 3 to 8. The lagged effects of
temperature were examined over 20 days and allowed 2, 4 or 6 degrees of freedom. The
Akaike Information Criterion (AIC) was used to judge the optimal degrees of freedom.

We tested for differences by age through studying the groups: all ages; ages 80 years;
and ages between 45–79 years.

We calculated the variation inflation factor as VIF = 1/(1–R-squared) to assess if
the variance estimation was inflated through multi-collinearity introduced by having
both temperature and heat wave terms in the same model. The variation inflation factor
(VIF) of the duration variable and temperature was estimated to 1.048, and for the
heat wave indicator variable to 1.058. So collinearity was not a concern for model
2 or 3.

Results

The AIC values associated with each model for mortality in all ages are in Table 3. Overall the models including additional heat wave variables (model 2 and 3) gave
a better fit to the data compared with model 1 for the same degrees of freedom. The
two best fits are a simple parameterization using an indicator variable or a heat
wave duration variable with 3 df for temperature and 2 df for the lagged effects as main effect. These models resulted in a small marginal
heat effect on non-heat wave days (Figure 2), while more flexible parameterizations for the main temperature effect estimated
a larger effect (Figures 2 and 3). However, with increasing complexity of the main temperature-mortality relationship
the model fit decreased and the bendiness of the estimated two dimensional spline
functions increased considerably as can be seen in Table 3 and Figures 2–4. Figures 2–4 also show that models including additional heat wave effects predict a lower mortality
increase for high temperatures at short lags compared with the model with no heat
wave variable. The effect of heat on lag 0 is then higher in the model without an
additional heat wave effect. This shows that the model without an additional heat
wave effect would estimate higher marginal excess mortality on days with high temperatures
that were not part of the heat wave compared to the model with the additional heat
wave effect. It appears that models with an explicit allowance for additional heat
wave effects do a better job of describing the exact distribution of heat-related
deaths.

Table 3. AIC values for the three models using 3 to 8 degrees of freedom for temperature and
2 to 6 degrees of freedom for lag, Stockholm, 1990–2002 in all ages

Figure 2.The distributed non-linear lag (2df)surface relating to a daily max temperature of 12 C. The models to the left have no heat wave variables (1), the models in the middle
include a heat wave duration variable (2), and the models to the right include a heat
wave indicator variable (3). The figure shows parameterizations using 2 df for the lags and 3, 6, and 8 df for the variable in the rows starting from the top respectively.

Figure 3.The distributed non-linear lag (4df)surface relating to a daily max temperature of 12 C. The models to the left are without additional heat wave variables (1), the models
in the middle include a heat wave duration variable (2), and the models to the right
include a heat wave indicator variable (3). The figure shows parameterizations using
4 df for the lags and 3, 6, and 8 df for the variable in the rows starting from the top respectively.

Figure 4.The distributed non-linear lag (6df)surface relating to a daily max temperature of 12 C. The models to the left are without additional heat wave variables (1), the models
in the middle include a heat wave duration variable (2), and the models to the right
include a heat wave indicator variable (3). The figure shows parameterizations using
6 df for the lags and 3, 6, and 8 df for the variable in the rows starting from the top respectively.

In the age group 80 years and above the inclusion of additional heat wave variables
did not improve the model fit (Table 4). In the age group 45–79 years, however, the inclusion of an additional heat wave
variable improved the model fit according to the AIC (Table 5), while it is not possible to distinguish between the model including the HWD (model
2) and HWI (model 3) in terms of model fit.

Table 4. AIC values for the three models using 3 to 8 degrees of freedom for temperature and
2 to 6 degrees of freedom for lag, Stockholm, 1990–2002 in ages 80 years of age and
above

Table 5. AIC values for the three models using 3 to 8 degrees of freedom for temperature and
2 to 6 degrees of freedom for lag, Stockholm, 1990–2002 in ages 45 to 79 years of
age

Figures 5 and 6 show the effects of a 5 unit increase in maximum daily temperature from 26 C to 31 C
(the heat wave threshold is 27.5 C). Figure 5 shows the excess mortality predictions for day 4 of a heat wave (including the 0–3 day
lagged effects) while varying the degrees of freedom of the main temperature-mortality
relationship, and showing the results for three different models. In broad terms,
the heat wave predictions are similar with more complex model fits, but models 2 and
3 (which include terms for heat wave effects) predict higher excess mortality for
the less complex temperature-mortality parameterizations. However, even for complex
parameterizations, the model without an additional heat wave effect predicts a few
percent lower excess mortality compared to the two heat wave models.

Figure 5.The excess mortality predictions for day 4 of a heat wave with maximum daily temperatures
at 31 C as compared to 2 C. The predictions include the 0–3 lagged effects over a range of different degrees
of freedom of the main temperature-mortality relationship and: i) from a model with
no added heat wave effect (model 1); ii) from a model with a heat wave duration parameter
(model 2); iii) and from a model with the added heat wave effect as a dummy variable
(model 3).

Figure 6.The excess mortality predictions for a single hot day (non-heat wave day) for maximum
daily temperatures at 31 C as compared to 26 C. The predictions include only the lag 0 effect over a range of different degrees of
freedom of the main temperature-mortality relationship and: i) from a model with no
added heat wave effect (model 1); ii) from a model with a heat wave duration parameter
(model 2); iii) and from a model with the added heat wave effect as a dummy variable
(model 3).

Figure 6 shows the corresponding marginal excess mortality predictions for a single hot day
(non-heat wave day; lag 0 only). The model without the added heat wave effect (model
1) predicts higher mortality on non-heat wave days compared to the models that differentiate
between heat wave and non-heat wave days (model 2 and 3). Thus, model 1 (without the
additional heat wave effect) may over-estimate the effect of heat on non-heat wave
days, while it appears to under-estimate the effect from heat on heat wave days. Models
2 or 3 do better in this sense, differentiating the effects between heat wave days
and non-heat wave days through the variables capturing the effect modification between
the lags of temperature. The more complex parameterizations of model 2 and 3 strongly
indicate over-fitting in graphical examinations, whilst having better AIC values.

The optimal fit of model 3 estimated a relative risk of 1.112 (95% confidence interval
of 1.050, 1.176) associated with heat wave days (95% confidence interval of 1.050,
1.176). The effects ranged between 1.116 and 1.083, and all estimates were statistically
significant at the 5% level as is shown Table 6.

We did not assess the estimates’ sensitivity to the parameterization (df) of long-term time trends, as this appears to have less influence on the heat wave
effect [12,19].

Discussion

Estimates of additional heat wave effects in models of temperature-related mortality
can be interpreted as a constrained form of non-linear effect-modifications between
lags of high temperature (or, similarly lag interactions). This can explain why such
effects have been found to significantly contribute to additional deaths during heat
waves in previous studies of temperature related mortality, over and above the effects
of temperature overall. This dispels the widespread belief that such effects are incorporated
through distributed lag models. From a mechanistic perspective including additional
effects from heat waves are supported through the physiological stress incurred by
cumulative exposure being potentially different from the stress from shorter periods
of extreme heat, and can also result in differences in the population at risk such
as contrasting susceptibility with age [8-10,13,19].

We found the additional heat wave effects were more important in middle age populations
and the elderly compared to the very elderly. We found the size of the heat wave effect
depended on the complexity of the main temperature-mortality parameterization, more
specifically on the df used for modeling non-linearity of temperature and lagged effects. This indicates
that there is, not surprisingly, some overlap between the effects of hot days and
the effects of heat waves. Our results show, however, that there is also likely to
be an independent extra effect of heat waves that is not captured by hot days, e.g.
an effect modification. Differences in the temperature-mortality parameterization
probably explain many of the differences between the conclusions of recent studies
on the effects of heat waves [6,8,9,11-13,19,21] In our example we found that a simple model for the temperature-mortality association
was better than a complex model. The additional heat wave effects are substantial
and important to account for in order not to underestimate mortality risks during
heat wave days. We conclude that it is important not to over fit the data by using
too complex non-linear lag parameterizations, and that simple parameters for the additional
effects of heat waves are useful. However, the more complex parameterizations appear
to better capture effects and the high end. It appear it may not be reasonable to
distribute the degrees of freedom uniformly over the temperature and lag scales as
such assumption gave rise to over-fitting in regions of the temperature and lag scale
where the relationship is not very complex. Overall, model fit improved as the complexity
of the model and flexibility of the splines were reduced. These simpler models also
had larger additional heat wave effects, as reported elsewhere [6,8,9,11-13,19,21].

Some studies have used two-stage models to describe the effects from temperature using
distributed lag non-linear models in the first stage and the additional effect associated
with heat waves in a second step [10,12]. We note, however, that this deviates from the conventional framework for modeling
of effect-modifications, and that it can potentially affect the estimates of additional
heat wave effects downward.

We achieved a better model fit using an indicator variable for heat waves rather than
the duration variable when studying all ages; nevertheless, models with the duration
variable performed better than the models without additional heat wave components,
and similarly well in the age group 45–79 years of age.

Conclusions

We conclude that it is important to continue to explore the magnitude of additional
heat wave effects in future studies of temperature-related mortality, e.g. temperature
lag effect-modifications. It appear also important to fit models that are location
sensitive in the parameterization (choice of df), as well as in the evaluation of potential additional heat wave effects. Fitting
a complex distributed lag non-linear model may reduce the heat wave signal and over-estimate
mortality on non-heat wave days, compared to a model including a heat wave term. It
is important to be able to differentiate between extended periods of heat and single
days of extremely high temperatures, since several recent studies have shown that
duration of heat exposure is related to mortality risk. Increasing the accuracy of
heat-wave mortality models will assist public health authorities to direct preventive
actions when they are most needed. Future studies should continue to study and identify
potential differences in the population at risk to heat and heat waves, as well as
describe the mechanistic differences.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

JR designed the study, analysed the data, interpreted the results and drafted the
manuscript. AB helped design the study, advised in the analysis of data, interpreted
the results and helped draft the paper. AW interpreted the results, advised in the
analysis and helped draft the paper.

Acknowledgements

Part of this work was undertaken within the Umeå Centre for Global Health Research
at Umeå University, with support from FAS, the Swedish Council for Working Life and
Social Research (grant no: 2006–1512).